Search

Let G be a finite abelian group and let exp (G) denote the least common multiple of the orders of all elements of G. A BH (G , h) matrix is a G-invariant | G | × | G | matrix H whose entries are complex hth roots of unity such that H H ∗ = | G | I | G | . By ν p (x) we denote the p-adic valuation of the integer x. Using bilinear forms over abelian groups, we [11] constructed new classes of BH (G , h) matrices under the following conditions. ν p (h) ≥ ⌈ ν p (exp (G)) / 2 ⌉ for any prime divisor p of |G|, and ν 2 (h) ≥ 2 if ν 2 (| G |) is odd and G has a direct factor Z 2 . The purpose of this paper is to further study the conditions on G and h so that a BH (G , h) matrix exists. We will focus on BH (Z n , h) and BH (G , 2 p b) matrices, where p is an odd prime. Our results describe various relation among |G|, gcd (| G | , h) and lcm (| G | , h) . Moreover, they confirm the nonexistence of 623 cases in the 3310 open cases for the existence of BH (Z n , h) matrices in which 1 ≤ n , h ≤ 100 . Finally, we show that BH (G , h) matrices can be used to construct a new family of perfect polyphase arrays. [ABSTRACT FROM AUTHOR]

Abstract Let p be an odd prime, let a be a positive integer, let m be an odd positive integer, and suppose that a generalized bent function from Z 2 p a m to Z 2 p a exists. We show that this implies m ≠ 1 , p ≤ 2 2 m + 2 m + 1 , and ord p (2) ≤ 2 m − 1. We obtain further necessary conditions and prove that p = 7 if m = 3 and p ∈ { 7 , 23 , 31 , 73 , 89 } if m = 5. Our results are based on new tools for the investigation of cyclotomic integers of prescribed complex modulus, including "minimal aliases" invariant under automorphisms, and bounds on the ℓ 2 -norms of their coefficient vectors. These methods have further applications, for instance, to relative difference sets, circulant Butson matrices, and other kinds of bent functions. [ABSTRACT FROM AUTHOR]

We investigate the existence problem of group invariant matrices using algebraic approaches. We extend the usual concept of multipliers to group rings with cyclotomic integers as coefficients. This concept is combined with the field descent method and rational idempotents to develop new non-existence results. [ABSTRACT FROM AUTHOR]

A finiteness phenomenon described by Loxton and later by Kedlaya states that, for any fixed m, there exist (modulo multiplication by roots of unity) only finitely many m-Weil numbers in ℚab. In the present paper we show that this phenomenon extends to all finite extensions of ℚab. [ABSTRACT FROM AUTHOR]

Abstract: We construct Weil numbers corresponding to genus-2 curves with p-rank 1 over the finite field of elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of -valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect to a subgroup of prescribed order. The curves are defined over out of necessity: we show that curves of p-rank 1 over for large p cannot be efficiently constructed using explicit CM constructions. [Copyright &y& Elsevier]

Abstract: Let C be a smooth curve over , being the valuation ring of an unramified extension of the field of p-adic numbers, with residue field . Let f be a function over C, and Ψ be an additive character of order over R; in this paper we study the exponential sums associated to f and Ψ over C, and their L-functions. We show the rationality of the L-functions in a more general setting, then in the case of curves we express them as products of L-functions associated to polynomials over the affine line, each factor coming from a singularity of f. Finally we show that in the case of Morse functions (i.e. having only simple singularities), the degree of the L-functions are, up to sign, the same as in the case of finite fields, yielding very similar bounds for exponential sums. [Copyright &y& Elsevier]

Abstract: The Leopoldt conjecture is concerned with the image of the global units in the local units at the primes dividing p. In the definition of the global units the infinite place is distinguished. Exchanging p and infinity in the formulation one gets a new conjecture. It predicts that certain vectors should be linearly independent over the reals whose components are arguments of conjugates of Weil numbers. Using Baker''s result on linear forms in logarithms we prove part of this new conjecture in certain abelian situations. [Copyright &y& Elsevier]